The definition of monotonicity seems to require the output to always increase for an increasing bit count. What about whether the output stays the same with increasing bit count? Some books and papers indicate that if the output stays the same, it will still be monotonic. Other books only mention it should always increase. What is the accepted industry standard for a converter to be monotonic?

I understand that monotonic increasing is equal as increasing or equal, as well as monotonic decreasing is equal as decreasing or equal. That is a thing that is better writing mathematically as speaking...

I think you are correct that it should apply for increasing or decreasing conditions.

Here is one of the definitions I found: "In a monotonic DAC, the analog output always increases or remains constant as the digital input increases. The analog output never decreases during the input sequence. If the analog output decreases at any point during the input sequence, a DAC is said to be non-monotonic."

Some textbooks and papers never mention the "or remains constant" part!

A monotonic DAC is one in which the analog output follows the direction of the digital input, no matter how the digital input varies. In other words, the analog output consistently increases as the input increases, and decreases as the input decreases. If the analog output were not to follow the direction of the digital input at any time — decrease when it increases or increase when it decreases — then the DAC would be considered non-monotonic.

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Sadly it's still ambiguous because one of the definitions of monotonicity is that the derivative is always positive (or negative) at any point, but if two samples are the same value then the derivative is zero, yet many definitions indicate greater than or equal to and not just greater than between samples.

At least, Analog Devices also consider the output staying the same as to be still monotonic.

From The AD569 data sheet::MONOTONICITY: A DAC is monotonic if the output eitherincreases or remains constant for increasing digital inputs. Allversions of the AD569 are monotonic over their full operatingtemperature range.

I think it is better to look not only DAC specifications but also application requirements those mandates DAC specifications.

A monotonic DAC is the one its output does not change direction while a min to max (or max to min) input code sweeping. As long as input code is changed only in one direction output also changes only one direction. Now of course output can stay on the same code for more than one input code and DAC is still monotonic -- i.e. output follows inputs direction and as long as there is no direction change stalling at one point is OK --. There are DNL and INL specifications ( DAC accuracy and linearity) those defines whether output is staying on the same code for changing input code and how accurate output steps per code. Because making a DNL(max) and INL(max) less or equal than 1LSB is harder and more expensive, manufacturers may opt to design a high resolution DAC with several LSB's of INL error as long as they can find customers to buy these DACs. These kinds of DAC's used in control system applications (not because a desired feature but does the job and cheaper). In control systems with feedback a DAC with monotonicity and high resolution is needed but linearity or absolute accuracy of DAC is not need to be good. For control systems higher resolution gives a better plant control/accuracy and as long as DAC is monotonic control system assures its control since there is a feedback that continuously correct for errors and DAC errors also corrected since DAC is within the control loop. If say an input code increase did not increase output code than since loop error is still not corrected, feedback simply requires input code to further increase to reduce error between input and output. Taking advantage of the feedback's watchfull eye and continuous control, a DAC in a control system could have several LSB of INL error since feedback system corrects for DAC non-linearity (DAC linearity not as important, i.e. adds a latency in control system, but DAC resolution is much important). And manufacturing high resolution DAC's with high linearity INL(max)=1LSB is much harder and expensive than manufacturing high resolution DAC with low linearity INL(max)> 1LSB. Sorry it turn out a long answer, but hope it helps.